# Frequency modulation

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In telecommunications and signal processing, frequency modulation (FM) is the encoding of information in a carrier wave by varying the instantaneous frequency of the wave.

Signal processing is a subfield of mathematics, information and electrical engineering that concerns the analysis, synthesis, and modification of signals, which are broadly defined as functions conveying "information about the behavior or attributes of some phenomenon", such as sound, images, and biological measurements. For example, signal processing techniques are used to improve signal transmission fidelity, storage efficiency, and subjective quality, and to emphasize or detect components of interest in a measured signal.

Information is the resolution of uncertainty; it is that which answers the question of "what an entity is" and is thus that which specifies the nature of that entity, as well as the essentiality of its properties. Information is associated with data and knowledge, as data is meaningful information and represents the values attributed to parameters, and knowledge signifies understanding of an abstract or concrete concept. The existence of information is uncoupled with an observer, which refers to that which accesses information to discern that which it specifies; information exists beyond an event horizon for example, and in the case of knowledge, the information itself requires a cognitive observer to be accessed.

In telecommunications, a carrier wave, carrier signal, or just carrier, is a waveform that is modulated (modified) with an input signal for the purpose of conveying information. This carrier wave usually has a much higher frequency than the input signal does. The purpose of the carrier is usually either to transmit the information through space as an electromagnetic wave, or to allow several carriers at different frequencies to share a common physical transmission medium by frequency division multiplexing. The term is also used for an unmodulated emission in the absence of any modulating signal.

## Contents

In analog frequency modulation, such as FM radio broadcasting of an audio signal representing voice or music, the instantaneous frequency deviation, the difference between the frequency of the carrier and its center frequency, is proportional to the modulating signal.

An analog signal is any continuous signal for which the time-varying feature (variable) of the signal is a representation of some other time varying quantity, i.e., analogous to another time varying signal. For example, in an analog audio signal, the instantaneous voltage of the signal varies continuously with the pressure of the sound waves. It differs from a digital signal, in which the continuous quantity is a representation of a sequence of discrete values which can only take on one of a finite number of values. The term analog signal usually refers to electrical signals; however, mechanical, pneumatic, hydraulic, human speech, and other systems may also convey or be considered analog signals.

Frequency deviation is used in FM radio to describe the maximum difference between an FM modulated frequency and the nominal carrier frequency. The term is sometimes mistakenly used as synonymous with frequency drift, which is an unintended offset of an oscillator from its nominal frequency.

Digital data can be encoded and transmitted via FM by shifting the carrier's frequency among a predefined set of frequencies representing digits – for example one frequency can represent a binary 1 and a second can represent binary 0. This modulation technique is known as frequency-shift keying (FSK). FSK is widely used in modems such as fax modems, and can also be used to send Morse code. [1] Radioteletype also uses FSK. [2]

Digital data, in information theory and information systems, is the discrete, discontinuous representation of information or works. Numbers and letters are commonly used representations.

In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: typically "0" (zero) and "1" (one).

Frequency-shift keying (FSK) is a frequency modulation scheme in which digital information is transmitted through discrete frequency changes of a carrier signal. The technology is used for communication systems such as telemetry, weather balloon radiosondes, caller ID, garage door openers, and low frequency radio transmission in the VLF and ELF bands. The simplest FSK is binary FSK (BFSK). BFSK uses a pair of discrete frequencies to transmit binary information. With this scheme, the "1" is called the mark frequency and the "0" is called the space frequency.

Frequency modulation is widely used for FM radio broadcasting. It is also used in telemetry, radar, seismic prospecting, and monitoring newborns for seizures via EEG, [3] two-way radio systems, music synthesis, magnetic tape-recording systems and some video-transmission systems. In radio transmission, an advantage of frequency modulation is that it has a larger signal-to-noise ratio and therefore rejects radio frequency interference better than an equal power amplitude modulation (AM) signal. For this reason, most music is broadcast over FM radio.

Telemetry is an automated communications process by which measurements and other data are collected at remote or inaccessible points and transmitted to receiving equipment for monitoring. The word is derived from Greek roots: tele = remote, and metron = measure. Systems that need external instructions and data to operate require the counterpart of telemetry, telecommand.

Frequency modulation and phase modulation are the two complementary principal methods of angle modulation; phase modulation is often used as an intermediate step to achieve frequency modulation. These methods contrast with amplitude modulation, in which the amplitude of the carrier wave varies, while the frequency and phase remain constant.

Phase modulation (PM) is a modulation pattern for conditioning communication signals for transmission. It encodes a message signal as variations in the instantaneous phase of a carrier wave. Phase modulation is one of the two principal forms of angle modulation, together with frequency modulation.

Angle modulation is a class of carrier modulation that is used in telecommunications transmission systems. The class comprises frequency modulation (FM) and phase modulation (PM), and is based on altering the frequency or the phase, respectively, of a carrier signal to encode the message signal. This contrasts with varying the amplitude of the carrier, practiced in amplitude modulation (AM) transmission, the earliest of the major modulation methods used widely in early radio broadcasting.

Amplitude modulation (AM) is a modulation technique used in electronic communication, most commonly for transmitting information via a radio carrier wave. In amplitude modulation, the amplitude of the carrier wave is varied in proportion to that of the message signal being transmitted. The message signal is, for example, a function of the sound to be reproduced by a loudspeaker, or the light intensity of pixels of a television screen. This technique contrasts with frequency modulation, in which the frequency of the carrier signal is varied, and phase modulation, in which its phase is varied.

## Theory

If the information to be transmitted (i.e., the baseband signal) is ${\displaystyle x_{m}(t)}$ and the sinusoidal carrier is ${\displaystyle x_{c}(t)=A_{c}\cos(2\pi f_{c}t)\,}$, where fc is the carrier's base frequency, and Ac is the carrier's amplitude, the modulator combines the carrier with the baseband data signal to get the transmitted signal:[ citation needed ]

{\displaystyle {\begin{aligned}y(t)&=A_{c}\cos \left(2\pi \int _{0}^{t}f(\tau )d\tau \right)\\&=A_{c}\cos \left(2\pi \int _{0}^{t}\left[f_{c}+f_{\Delta }x_{m}(\tau )\right]d\tau \right)\\&=A_{c}\cos \left(2\pi f_{c}t+2\pi f_{\Delta }\int _{0}^{t}x_{m}(\tau )d\tau \right)\\\end{aligned}}}

where ${\displaystyle f_{\Delta }=K_{f}A_{m}}$, ${\displaystyle K_{f}}$ being the sensitivity of the frequency modulator and ${\displaystyle A_{m}}$ being the amplitude of the modulating signal or baseband signal.

In this equation, ${\displaystyle f(\tau )\,}$ is the instantaneous frequency of the oscillator and ${\displaystyle f_{\Delta }\,}$ is the frequency deviation , which represents the maximum shift away from fc in one direction, assuming xm(t) is limited to the range ±1.

While most of the energy of the signal is contained within fc ± fΔ, it can be shown by Fourier analysis that a wider range of frequencies is required to precisely represent an FM signal. The frequency spectrum of an actual FM signal has components extending infinitely, although their amplitude decreases and higher-order components are often neglected in practical design problems. [4]

### Sinusoidal baseband signal

Mathematically, a baseband modulating signal may be approximated by a sinusoidal continuous wave signal with a frequency fm. This method is also named as single-tone modulation. The integral of such a signal is:

${\displaystyle \int _{0}^{t}x_{m}(\tau )d\tau =A_{m}{\frac {\sin \left(2\pi f_{m}t\right)}{2\pi f_{m}}}\,}$

In this case, the expression for y(t) above simplifies to:

${\displaystyle y(t)=A_{c}\cos \left(2\pi f_{c}t+{\frac {A_{m}f_{\Delta }}{f_{m}}}\sin \left(2\pi f_{m}t\right)\right)\,}$

where the amplitude ${\displaystyle A_{m}\,}$ of the modulating sinusoid is represented by the peak deviation ${\displaystyle f_{\Delta }\,}$ (see frequency deviation).

The harmonic distribution of a sine wave carrier modulated by such a sinusoidal signal can be represented with Bessel functions; this provides the basis for a mathematical understanding of frequency modulation in the frequency domain.

### Modulation index

As in other modulation systems, the modulation index indicates by how much the modulated variable varies around its unmodulated level. It relates to variations in the carrier frequency:

${\displaystyle h={\frac {\Delta {}f}{f_{m}}}={\frac {f_{\Delta }\left|x_{m}(t)\right|}{f_{m}}}}$

where ${\displaystyle f_{m}\,}$ is the highest frequency component present in the modulating signal xm(t), and ${\displaystyle \Delta {}f\,}$ is the peak frequency-deviation—i.e. the maximum deviation of the instantaneous frequency from the carrier frequency. For a sine wave modulation, the modulation index is seen to be the ratio of the peak frequency deviation of the carrier wave to the frequency of the modulating sine wave.

If ${\displaystyle h\ll 1}$, the modulation is called narrowband FM (NFM), and its bandwidth is approximately ${\displaystyle 2f_{m}\,}$. Sometimes modulation index ${\displaystyle h<0.3}$ is considered as NFM, otherwise wideband FM (WFM or FM).

For digital modulation systems, for example binary frequency shift keying (BFSK), where a binary signal modulates the carrier, the modulation index is given by:

${\displaystyle h={\frac {\Delta {}f}{f_{m}}}={\frac {\Delta {}f}{\frac {1}{2T_{s}}}}=2\Delta {}fT_{s}\ }$

where ${\displaystyle T_{s}\,}$ is the symbol period, and ${\displaystyle f_{m}={\frac {1}{2T_{s}}}\,}$ is used as the highest frequency of the modulating binary waveform by convention, even though it would be more accurate to say it is the highest fundamental of the modulating binary waveform. In the case of digital modulation, the carrier ${\displaystyle f_{c}\,}$ is never transmitted. Rather, one of two frequencies is transmitted, either ${\displaystyle f_{c}+\Delta {}f}$ or ${\displaystyle f_{c}-\Delta {}f}$, depending on the binary state 0 or 1 of the modulation signal.

If ${\displaystyle h\gg 1}$, the modulation is called wideband FM and its bandwidth is approximately ${\displaystyle 2f_{\Delta }\,}$. While wideband FM uses more bandwidth, it can improve the signal-to-noise ratio significantly; for example, doubling the value of ${\displaystyle \Delta {}f\,}$, while keeping ${\displaystyle f_{m}}$ constant, results in an eight-fold improvement in the signal-to-noise ratio. [5] (Compare this with chirp spread spectrum, which uses extremely wide frequency deviations to achieve processing gains comparable to traditional, better-known spread-spectrum modes).

With a tone-modulated FM wave, if the modulation frequency is held constant and the modulation index is increased, the (non-negligible) bandwidth of the FM signal increases but the spacing between spectra remains the same; some spectral components decrease in strength as others increase. If the frequency deviation is held constant and the modulation frequency increased, the spacing between spectra increases.

Frequency modulation can be classified as narrowband if the change in the carrier frequency is about the same as the signal frequency, or as wideband if the change in the carrier frequency is much higher (modulation index > 1) than the signal frequency. [6] For example, narrowband FM (NFM) is used for two-way radio systems such as Family Radio Service, in which the carrier is allowed to deviate only 2.5 kHz above and below the center frequency with speech signals of no more than 3.5 kHz bandwidth. Wideband FM is used for FM broadcasting, in which music and speech are transmitted with up to 75 kHz deviation from the center frequency and carry audio with up to a 20 kHz bandwidth and subcarriers up to 92 kHz.

### Bessel functions

For the case of a carrier modulated by a single sine wave, the resulting frequency spectrum can be calculated using Bessel functions of the first kind, as a function of the sideband number and the modulation index. The carrier and sideband amplitudes are illustrated for different modulation indices of FM signals. For particular values of the modulation index, the carrier amplitude becomes zero and all the signal power is in the sidebands. [4]

Since the sidebands are on both sides of the carrier, their count is doubled, and then multiplied by the modulating frequency to find the bandwidth. For example, 3 kHz deviation modulated by a 2.2 kHz audio tone produces a modulation index of 1.36. Suppose that we limit ourselves to only those sidebands that have a relative amplitude of at least 0.01. Then, examining the chart shows this modulation index will produce three sidebands. These three sidebands, when doubled, gives us (6 × 2.2 kHz) or a 13.2 kHz required bandwidth.

Modulation
index
Sideband amplitude
Carrier12345678910111213141516
0.001.00
0.250.980.12
0.50.940.240.03
1.00.770.440.110.02
1.50.510.560.230.060.01
2.00.220.580.350.130.03
2.410.000.520.430.200.060.02
2.5−0.050.500.450.220.070.020.01
3.0−0.260.340.490.310.130.040.01
4.0−0.40−0.070.360.430.280.130.050.02
5.0−0.18−0.330.050.360.390.260.130.050.02
5.530.00−0.34−0.130.250.400.320.190.090.030.01
6.00.15−0.28−0.240.110.360.360.250.130.060.02
7.00.300.00−0.30−0.170.160.350.340.230.130.060.02
8.00.170.23−0.11−0.29−0.100.190.340.320.220.130.060.03
8.650.000.270.06−0.24−0.230.030.260.340.280.180.100.050.02
9.0−0.090.250.14−0.18−0.27−0.060.200.330.310.210.120.060.030.01
10.0−0.250.040.250.06−0.22−0.23−0.010.220.320.290.210.120.060.030.01
12.00.05−0.22−0.080.200.18−0.07−0.24−0.170.050.230.300.270.200.120.070.030.01

### Carson's rule

A rule of thumb, Carson's rule states that nearly all (~98 percent) of the power of a frequency-modulated signal lies within a bandwidth ${\displaystyle B_{T}\,}$ of:

${\displaystyle B_{T}=2\left(\Delta f+f_{m}\right)=2f_{m}(\beta +1)}$

where ${\displaystyle \Delta f\,}$, as defined above, is the peak deviation of the instantaneous frequency ${\displaystyle f(t)\,}$ from the center carrier frequency ${\displaystyle f_{c}}$, ${\displaystyle \beta }$ is the Modulation index which is the ratio of frequency deviation to highest frequency in the modulating signal and ${\displaystyle f_{m}\,}$is the highest frequency in the modulating signal. Condition for application of Carson's rule is only sinusoidal signals.

${\displaystyle B_{T}=2(\Delta f+W)=2W(D+1)}$

where W is the highest frequency in the modulating signal but non-sinusoidal in nature and D is the Deviation ratio which the ratio of frequency deviation to highest frequency of modulating non-sinusoidal signal.

## Noise reduction

FM provides improved signal-to-noise ratio (SNR), as compared for example with AM. Compared with an optimum AM scheme, FM typically has poorer SNR below a certain signal level called the noise threshold, but above a higher level – the full improvement or full quieting threshold – the SNR is much improved over AM. The improvement depends on modulation level and deviation. For typical voice communications channels, improvements are typically 5–15 dB. FM broadcasting using wider deviation can achieve even greater improvements. Additional techniques, such as pre-emphasis of higher audio frequencies with corresponding de-emphasis in the receiver, are generally used to improve overall SNR in FM circuits. Since FM signals have constant amplitude, FM receivers normally have limiters that remove AM noise, further improving SNR. [7] [8]

## Implementation

### Modulation

FM signals can be generated using either direct or indirect frequency modulation:

### Demodulation

Many FM detector circuits exist. A common method for recovering the information signal is through a Foster-Seeley discriminator or ratio detector. A phase-locked loop can be used as an FM demodulator. Slope detection demodulates an FM signal by using a tuned circuit which has its resonant frequency slightly offset from the carrier. As the frequency rises and falls the tuned circuit provides a changing amplitude of response, converting FM to AM. AM receivers may detect some FM transmissions by this means, although it does not provide an efficient means of detection for FM broadcasts.

## Applications

### Magnetic tape storage

FM is also used at intermediate frequencies by analog VCR systems (including VHS) to record the luminance (black and white) portions of the video signal. Commonly, the chrominance component is recorded as a conventional AM signal, using the higher-frequency FM signal as bias. FM is the only feasible method of recording the luminance ("black and white") component of video to (and retrieving video from) magnetic tape without distortion; video signals have a large range of frequency components – from a few hertz to several megahertz, too wide for equalizers to work with due to electronic noise below −60  dB. FM also keeps the tape at saturation level, acting as a form of noise reduction; a limiter can mask variations in playback output, and the FM capture effect removes print-through and pre-echo. A continuous pilot-tone, if added to the signal – as was done on V2000 and many Hi-band formats – can keep mechanical jitter under control and assist timebase correction.

These FM systems are unusual, in that they have a ratio of carrier to maximum modulation frequency of less than two; contrast this with FM audio broadcasting, where the ratio is around 10,000. Consider, for example, a 6-MHz carrier modulated at a 3.5-MHz rate; by Bessel analysis, the first sidebands are on 9.5 and 2.5 MHz and the second sidebands are on 13 MHz and −1 MHz. The result is a reversed-phase sideband on +1 MHz; on demodulation, this results in unwanted output at 6 – 1 = 5 MHz. The system must be designed so that this unwanted output is reduced to an acceptable level. [10]

### Sound

FM is also used at audio frequencies to synthesize sound. This technique, known as FM synthesis, was popularized by early digital synthesizers and became a standard feature in several generations of personal computer sound cards.

Edwin Howard Armstrong (1890–1954) was an American electrical engineer who invented wideband frequency modulation (FM) radio. [11] He patented the regenerative circuit in 1914, the superheterodyne receiver in 1918 and the super-regenerative circuit in 1922. [12] Armstrong presented his paper, "A Method of Reducing Disturbances in Radio Signaling by a System of Frequency Modulation", (which first described FM radio) before the New York section of the Institute of Radio Engineers on November 6, 1935. The paper was published in 1936. [13]

An FM signal can also be used to carry a stereo signal; this is done with multiplexing and demultiplexing before and after the FM process. The FM modulation and demodulation process is identical in stereo and monaural processes. A high-efficiency radio-frequency switching amplifier can be used to transmit FM signals (and other constant-amplitude signals). For a given signal strength (measured at the receiver antenna), switching amplifiers use less battery power and typically cost less than a linear amplifier. This gives FM another advantage over other modulation methods requiring linear amplifiers, such as AM and QAM.

FM is commonly used at VHF radio frequencies for high-fidelity broadcasts of music and speech. Analog TV sound is also broadcast using FM. Narrowband FM is used for voice communications in commercial and amateur radio settings. In broadcast services, where audio fidelity is important, wideband FM is generally used. In two-way radio, narrowband FM (NBFM) is used to conserve bandwidth for land mobile, marine mobile and other radio services.

There are reports that on October 5, 1924, Professor Mikhail A. Bonch-Bruevich, during a scientific and technical conversation in the Nizhny Novgorod Radio Laboratory, reported about his new method of telephony, based on a change in the period of oscillations. Demonstration of frequency modulation was carried out on the laboratory model. [14]

## Related Research Articles

In electronics and telecommunications, modulation is the process of varying one or more properties of a periodic waveform, called the carrier signal, with a modulating signal that typically contains information to be transmitted. Most radio systems in the 20th century used frequency modulation (FM) or amplitude modulation (AM) for radio broadcast.

In telecommunications, orthogonal frequency-division multiplexing (OFDM) is a method of encoding digital data on multiple carrier frequencies. OFDM has developed into a popular scheme for wideband digital communication, used in applications such as digital television and audio broadcasting, DSL internet access, wireless networks, power line networks, and 4G mobile communications.

In radio communications, single-sideband modulation (SSB) or single-sideband suppressed-carrier modulation (SSB-SC) is a type of modulation, used to transmit information, such as an audio signal, by radio waves. A refinement of amplitude modulation, it uses transmitter power and bandwidth more efficiently. Amplitude modulation produces an output signal the bandwidth of which is twice the maximum frequency of the original baseband signal. Single-sideband modulation avoids this bandwidth increase, and the power wasted on a carrier, at the cost of increased device complexity and more difficult tuning at the receiver.

Baseband is a signal that has a near-zero frequency range, i.e. a spectral magnitude that is nonzero only for frequencies in the vicinity of the origin and negligible elsewhere. In telecommunications and signal processing, baseband signals are transmitted without modulation, that is, without any shift in the range of frequencies of the signal. Baseband has a low-frequency—contained within the bandwidth frequency close to 0 hertz up to a higher cut-off frequency. Baseband can be synonymous with lowpass or non-modulated, and is differentiated from passband, bandpass, carrier-modulated, intermediate frequency, or radio frequency (RF).

In telecommunication, Carson's bandwidth rule defines the approximate bandwidth requirements of communications system components for a carrier signal that is frequency modulated by a continuous or broad spectrum of frequencies rather than a single frequency. Carson's rule does not apply well when the modulating signal contains discontinuities, such as a square wave. Carson's rule originates from John Renshaw Carson's 1922 paper.

Double-sideband suppressed-carrier transmission (DSB-SC) is transmission in which frequencies produced by amplitude modulation (AM) are symmetrically spaced above and below the carrier frequency and the carrier level is reduced to the lowest practical level, ideally being completely suppressed.

In radio communications, a sideband is a band of frequencies higher than or lower than the carrier frequency, containing power as a result of the modulation process. The sidebands carry the information (modulation) transmitted by the signal. The sidebands consist of all the Fourier components of the modulated signal except the carrier. All forms of modulation produce sidebands.

Very High Frequency (VHF) Omni-Directional Range (VOR) is a type of short-range radio navigation system for aircraft, enabling aircraft with a receiving unit to determine its position and stay on course by receiving radio signals transmitted by a network of fixed ground radio beacons. It uses frequencies in the very high frequency (VHF) band from 108.00 to 117.95 MHz. Developed in the United States beginning in 1937 and deployed by 1946, VOR is the standard air navigational system in the world, used by both commercial and general aviation. By 2000 there were about 3,000 VOR stations around the world including 1,033 in the US, reduced to 967 by 2013 with more stations being decommissioned with the widespread adoption of GPS.

The short-time Fourier transform (STFT), is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. In practice, the procedure for computing STFTs is to divide a longer time signal into shorter segments of equal length and then compute the Fourier transform separately on each shorter segment. This reveals the Fourier spectrum on each shorter segment. One then usually plots the changing spectra as a function of time.

In pulsed radar and sonar signal processing, an ambiguity function is a two-dimensional function of time delay and Doppler frequency showing the distortion of a returned pulse due to the receiver matched filter due to the Doppler shift of the return from a moving target. The ambiguity function is determined by the properties of the pulse and the matched filter, and not any particular target scenario. Many definitions of the ambiguity function exist; Some are restricted to narrowband signals and others are suitable to describe the propagation delay and Doppler relationship of wideband signals. Often the definition of the ambiguity function is given as the magnitude squared of other definitions (Weiss). For a given complex baseband pulse , the narrowband ambiguity function is given by

Continuous-wave radar is a type of radar system where a known stable frequency continuous wave radio energy is transmitted and then received from any reflecting objects. Continuous-wave (CW) radar uses Doppler, which renders the radar immune to interference from large stationary objects and slow moving clutter.

In digital modulation, minimum-shift keying (MSK) is a type of continuous-phase frequency-shift keying that was developed in the late 1950s and 1960s. Similar to OQPSK, MSK is encoded with bits alternating between quadrature components, with the Q component delayed by half the symbol period.

A damped wave is a wave whose amplitude of oscillation decreases with time, eventually going to zero, an exponentially decaying sinusoidal wave. This term also refers to an early method of radio transmission produced by the first radio transmitters, spark gap transmitters, which consisted of a series of damped radio waves. Information was carried on this signal by telegraphy, turning the transmitter on and off to send messages in Morse code. Damped waves were the first practical means of radio communication, used during the wireless telegraphy era which ended around 1920. In radio engineering it is now generally referred to as "Class B" emission. However, such transmissions have a wide bandwidth and generate electrical "noise" which interferes with other radio transmissions.

Radar engineering details are technical details pertaining to the components of a radar and their ability to detect the return energy from moving scatterers — determining an object's position or obstruction in the environment. This includes field of view in terms of solid angle and maximum unambiguous range and velocity, as well as angular, range and velocity resolution. Radar sensors are classified by application, architecture, radar mode, platform, and propagation window.

In 1933, Edwin H. Armstrong patented a method for generating frequency modulation of radio signals. The Armstrong method generates a double sideband suppressed carrier signal, phase shifts this signal, and then reinserts the carrier to produce a frequency modulated signal.

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